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  • ** [[Elementary symmetric polynomial]] * [[Sequence | Sequences]] and [[Series]]
    2 KB (198 words) - 15:06, 7 December 2024
  • ...give the terms of a [[sequence]] which is of interest. Therefore the power series (i.e. the generating function) is <math>c_0 + c_1 x + c_2 x^2 + \cdots </ma The reason to go to such lengths is that our above polynomial is equal to <math>(1+x)^n</math> (which is clearly seen due to the [[Binomi
    4 KB (659 words) - 11:54, 7 March 2022
  • ...arithmetic sequence. All infinite arithmetic series diverge. As for finite series, there are two primary formulas used to compute their value. The first is that if an arithmetic series has first term <math>a_1</math>, last term <math>a_n</math>, and <math>n</m
    4 KB (736 words) - 01:00, 7 March 2024
  • ...10 times the sum of the original series. The common ratio of the original series is <math> \frac mn </math> where <math> m </math> and <math> n </math> are Let <math> P(x) </math> be a polynomial with integer coefficients that satisfies <math> P(17)=10 </math> and <math>
    7 KB (1,119 words) - 20:12, 28 February 2020
  • The polynomial <math> P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17} </math> has <math>34</math> co ...ctly, but there is one obvious transformation to make: sum the [[geometric series]]:
    2 KB (298 words) - 19:02, 4 July 2013
  • The [[polynomial]] <math>1-x+x^2-x^3+\cdots+x^{16}-x^{17}</math> may be written in the form Using the [[geometric series]] formula, <math>1 - x + x^2 + \cdots - x^{17} = \frac {1 - x^{18}}{1 + x}
    6 KB (872 words) - 15:51, 9 June 2023
  • We start off by adding <math>z^5</math> to both sides, to get a neat geometric sequence with <math>a = 1</math> and <math>r = z</math>, which gives us <ma ...hat <math>(x^6+x^3+1)+(x^4+x^2+1)=1</math>. Then, using sum of a geometric series, <math>\frac{x^9-1}{x^3-1}+\frac{x^6-1}{x^2-1}=1</math>.
    6 KB (1,064 words) - 22:24, 28 September 2024
  • ...>2\sqrt{10}(t^3 + 3t) = 200x^3 - \frac{2}{10x^3}</math>, which reduces the polynomial to just <math>(t^2 + 3)\left(2\sqrt{10}t + 1\right) = 0</math>. Then one ca ...we make the substitution, <math>x = -\frac{i}{\sqrt{10}}y</math>. The the polynomial becomes
    6 KB (1,060 words) - 16:36, 26 April 2024
  • A '''series''' is a sum of consecutive terms in a [[sequence]]. Common series are based on common sequences. ==Common Series==
    400 bytes (43 words) - 20:21, 22 July 2021
  • Compute the sum of all twenty-one terms of the geometric series ...and <math>n</math> are positive integers such that <math>m < n</math>, the geometric mean of <math>m</math> and <math>n</math> is greater than <math>2007</math>
    30 KB (4,794 words) - 22:00, 8 May 2024
  • By adding the same constant to <math>20,50,100</math> a geometric progression results. The common ratio is: Given the polynomial <math>a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n</math>, where <math>n</math> is
    22 KB (3,348 words) - 11:53, 22 July 2024
  • ...presented his proof of the Poincaré Conjecture during the Simons Lecture Series at the MIT Mathematics Department. He gave three lectures, titled "Ricci Fl ...uctures known as [[Hodge class]]es, which can be elementarily described as geometric representations of a given [[manifold]]'s topological properties, are compo
    13 KB (1,969 words) - 16:57, 22 February 2024
  • geometric progression. Find the sum of the lengths in cm of all the edges of this sol .... Thinking she knows everything there is about how to work with arithmetic series, she nearly turns right around to walk back home when Dr. Lisi poses a more
    71 KB (11,749 words) - 11:39, 20 November 2024
  • The '''characteristic polynomial''' of a linear [[operator]] refers to the [[polynomial]] whose roots are the [[eigenvalue]]s of the operator. It carries much info In the context of problem-solving, the characteristic polynomial is often used to find closed forms for the solutions of [[#Linear recurrenc
    19 KB (3,412 words) - 13:57, 21 September 2022
  • ...math> Now, we can say that <cmath>P(m) = (m-1)Q(m) - 2010</cmath> for some polynomial <math>Q(m)</math> with integer coefficients. Then if <math>P(m) = 0</math>, ...s sum to a power of <math>m</math>, we realize that the sum is a geometric series:
    11 KB (1,811 words) - 19:45, 1 September 2024
  • This is an infinite [[geometric series]] with first term <math>\frac{1}{81}</math> and common ratio <math>\frac{1} ...c{1}{9^{n+1}} \). Then this problem asks us to find the sum of a geometric series with first term 181 and common ratio 19.
    6 KB (900 words) - 00:16, 25 November 2024
  • Let <math>p(x)</math> be the polynomial <math>(1-x)^a(1-x^2)^b(1-x^3)^c\cdots(1-x^{32})^k</math>, where <math>a, b, ...everse the order of the coefficients of each factor, then we will obtain a polynomial whose coefficients are exactly the coefficients of <math>p(x)</math> in rev
    8 KB (1,348 words) - 08:44, 25 June 2022
  • The coefficient of <math> x^7 </math> in the polynomial expansion of ...terms is <math> 9 </math>. What is the sum of the first five terms of this series?
    15 KB (2,151 words) - 13:04, 19 February 2020
  • The polynomial <math>(x+y)^9</math> is expanded in decreasing powers of <math>x</math>. Th If the first term of an infinite geometric series is a positive integer, the common ratio is the reciprocal of a positive int
    16 KB (2,513 words) - 18:40, 5 September 2024
  • ...<math>-1<r<1</math>, let <math>S(r)</math> denote the sum of the geometric series <cmath>12+12r+12r^2+12r^3+\cdots .</cmath> Let <math>a</math> between <mat ...uence <math>a_{2k-1}</math>, <math>a_{2k}</math>, <math>a_{2k+1}</math> is geometric and the subsequence <math>a_{2k}</math>, <math>a_{2k+1}</math>, <math>a_{2k
    8 KB (1,360 words) - 00:05, 29 November 2024

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